DiffEqs`DETools`

This package provides the following functions for solving differential equations.

FOLDSolve[eqn, y[t], t]     solves a first-order linear differential equation for y[t].      BernoulliDSolve[eqn, y[t], t]     solves a Bernoulli-type differential equation for y[t].      RiccatiDSolve[eqn, y[t], t, z[t]]     solves a Riccati-type differential equation for y[t], given a known solution z[t].      SeparableDSolve[]     returns an implicit solution of a separable differential equation. SeriesDSolve[eqn, y[t], t, t0 , n]     returns the first n terms of a series expansion about t0 of the solution of a first- or second-order differential equation.

Finding solutions of differential equations.

This loads the package.

FOLDSolve

A few examples of the use of FOLDSolve:

Here’s an example on which DSolve works for a very long time.

BernoulliDSolve

A few examples of the use of BernoulliDSolve:

RiccatiDSolve

A few examples of the use of RiccatiDSolve:

The fourth argument is a known solution.

Here’s an example on which DSolve fails.

SeparableDSolve

A few examples of the use of SeparableDSolve:

The equation is given in separated form.

The next example uses the InitialPoint option. Note that the first number is a value for the variable on the left side of the equation and the second number is a value for the variable on the right side of the equation. Here the initial condition is that y = 0 when t = 1.

Since SeparableDSolve returns an equation, it can be combined easily with Solve.

A result from SeparableDSolve is easy to feed into PlotImplicit from the DEGraphics package.

SeriesDSolve

A few examples of the use of SeriesDSolve:

A nonlinear first-order initial-value problem:

A nonlinear second-order initial-value problem:

For a linear second-order equation (with no initial conditions), SeriesDSolve returns partial sums of a pair of “fundamental” solutions, if possible.

Here we get expansions about a point other than 0.

Nonpolynomial coefficients are no problem.

If there is a nonhomogeneous term,  SeriesDSolve returns the “rest” solution (i.e., the solution satisfying zero initial conditions).

Complex powers and Frobenius-series solutions are found as well.